Ion crystal size and structure in Paul traps

This article presents a relationship between the size of an ion crystal in a Paul trap and the radio frequency potential applied on the central ring electrode. Using a simple and elegant derivation it has been shown that the distance of an ion in the crystal from the trap center is proportional to <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mo>-</mml:mo><mml:mn>2</mml:mn><mml:mo>/</mml:mo><mml:mn>3</mml:mn></mml:math> power of the applied radio frequency voltage. The validity of this power law has been demonstrated on ion crystals having up to 13 ions. A spring-mass model has been presented to predict structure of ion crystals in Paul traps operating in the Dehmelt regime. Structures are obtained by minimizing the total potential energy stored in an ion ensemble. The total potential energy is taken to be the sum of the electrostatic potential due to ion-ion interaction and the potential energy stored in the springs. Structures of crystals having up to 13 ions predicted by our model have been verified by comparing them with results obtained by direct numerical simulations.